Splitting of Gysin extensions

A. J. Berrick, A. A. Davydov

Abstract.
Let X --> B be an orientable sphere bundle. Its Gysin sequence
exhibits H^*(X) as an extension of H^*(B)-modules. We prove that the
class of this extension is the image of a canonical class that we
define in the Hochschild 3-cohomology of H^*(B), corresponding to a
component of its A_infty-structure, and generalizing the Massey triple
product. We identify two cases where this class vanishes, so that the
Gysin extension is split. The first, with rational coefficients, is
that where B is a formal space; the second, with integer coefficients,
is where B is a torus.